The LDDMM optimal control problem has interesting developments when additional constraints are applied to the evolving diffeomorphism. Some examples, with applications to curve and surface matching, are developed in [1-2], where the notion of "Gaussian diffeons" is introduced. Such diffeons are vector fields shaped like a possibly skewed Gaussian, who generate a diffeomorphism and are reshaped and advected by it.

• [1] Constrained Diffeomorphic Shape Evolution, L. Younes, Foundations of Computational Mathematics 12 (3), 295-325, 2012
• [2] Gaussian diffeons for surface and image matching within a Lagrangian framework, L. Younes, Geometry, Imaging and Computing, 1 (1) pp. 141-171, 2014

A comprehensive and general discussion of the constrained setting has been developed in
[3] Shape deformation analysis from the optimal control viewpoint, S Arguillère, E Trélat, A Trouvé, L Younes, Journal de mathématiiques pures et appliquées, 104, 1, pp139-178, 2015. [4] Shape deformation and Optimal Control, S Arguillère, E Trélat, A Trouvé, L Younes, ESAIM:ProcS, 45, 300-307, 2013.

Other work using constrained LDDMM in specific contexts can be found in:
[5] Diffeomorphic Surface Registration with Atrophy Constraints, SIAM J. Imaging Sci., 9(3), 975–1003 (2016)
[6] Registration of Multiple Shapes using Constrained Optimal Control SIAM J. Imaging Sci., 9(1), 344–385, 2016
[7] Sub-Riemannian Methods in Shape Analysis, L Younes, B Gris, A Trouvé, Handbook of Variational Methods for Nonlinear Geometric Data, 463-495, 2020